L2-energy Method Research Articles

Large-time behavior of the solutions for an inflow problem of the full compressible Navier–Stokes–Korteweg equations

In this article, we are concerned with the large-time behavior of solutions to an inflow problem for the one-dimensional full compressible Navier–Stokes–Korteweg equations, which can be used to model the compressible viscous fluids with internal capillarity and heat conductivity. Under some suitable assumptions of the far fields and boundary values of the density, the velocity and the absolute temperature, the large-time behavior of the solutions for the initial boundary value problem is govern by the stationary solution and basic waves of the corresponding hyperbolic system. Then the asymptotic stability of the boundary layer, and the superposition of the boundary layer and the three-rarefaction wave are shown provided that the initial perturbation and the strength of the rarefaction wave and stationary wave are small. The proof is mainly based on L2-energy method, some time-decay estimates in Lp-norm for the smoothed rarefaction wave and the space variable decay estimates of the boundary layer. This can be viewed as the first result about the nonlinear stability of the combination of two different wave patterns for the inflow problem of the full compressible Navier–Stokes–Korteweg equations.

Convergence rates to the planar stationary solution to a 2D model of the radiating gas on half space

This paper is concerned with the asymptotic stability of a planar stationary solution to an initial-boundary value problem for a two-dimensional hyperbolic–elliptic coupled system of the radiating gas on half space. We show that the solution to the problem converges to the corresponding planar stationary solution as time tends to infinity under small initial perturbation. This result is proved by the standard L2-energy method and the div–curl decomposition. Moreover, we prove that the solution (u, q) converges to the corresponding planar stationary solution at the rate t−α/2−1/4 for the non-degenerate case and t−1/4 for the degenerate case. The proof is based on the time and space weighted energy method.

Stability of stationary solutions and viscous shock wave in the inflow problem for isentropic Navier-Stokes-Korteweg system

In this paper, we are concerned with the inflow problem in the half line (0,∞) to the isentropic compressible Navier-Stokes-Korteweg system. We first show existence and asymptotic stability of the stationary solutions to the inflow problem. Also, for shock wave, asymptotic profile of the inflow problem is shown to be a shifted viscous shock profile, which is suitably away from the boundary, and we prove that if the initial data around the shifted viscous shock profile and the strength of shifted viscous shock profile are sufficiently small, then the inflow problem has a unique global strong solution, which tends to the shifted viscous shock profile as time goes to infinity. Moreover, we show the stability of superposition viscous shock profiles and stationary solutions. The analysis is based on the elementary L2-energy method, but various techniques are introduced to establish the uniform boundary and energy estimates.

Asymptotic stability of a composite wave for the one-dimensional compressible micropolar fluid model without viscosity

We are concerned with the large time behavior of solutions to the Cauchy problem of the one-dimensional compressible micropolar fluid model without viscosity, where the far-field states of the initial data are prescribed to be different. If the corresponding Riemann problem of the compressible Euler system admits a contact discontinuity and two rarefaction waves solutions, we show that for such a non-viscous model, the combination of the viscous contact wave with two rarefaction waves is time-asymptotically stable provided that the strength of the composite wave and the initial perturbation are sufficiently small. The proof is given by an elementary L2 energy method.

The relativistic Boltzmann equation for soft potentials

The paper concerns the Cauchy problem on the relativistic Boltzmann equation for soft potentials in a periodic box. We show that the global-in-time solutions around relativistic Maxwellians exist in the weighted L∞ perturbation framework and also approach equilibrium states in large time in the weighted L2 framework at the rate of exp⁡(−λtβ) for some λ>0 and β∈(0,1). The proof is based on the nonlinear L2 energy method and nonlinear L∞ pointwise estimate with appropriate exponential weights in momentum. The results extend those on the classical Boltzmann equation by Caflisch [2,3] and Strain and Guo [31] to the relativistic version, and also improve the recent result on almost exponential time-decay by Strain [28] to the exponential rate. Moreover, we study the propagation of spatial regularity for the obtained solutions and also the large time behavior in the corresponding regular Sobolev space, provided that the spatial derivatives of initial data are bounded, not necessarily small.

Asymptotic behavior of radially symmetric solutions for a quasilinear hyperbolic fluid model in higher dimensions

We consider the large time behavior of the radially symmetric solution to the equation for a quasilinear hyperbolic model in the exterior domain of a ball in general space dimensions. In the previous paper [2], we proved the asymptotic stability of the stationary wave of the Burgers equations in the same exterior domain when the solution is also radially symmetric. On the other hand, in the 1D-case, a similar asymptotic structure as above to the damped wave equation with a convection term has been established by Ueda [10] and Ueda–Kawashima [11]. Assuming a certain condition on the boundary data on the ball and the behavior at infinity of the fluid, we shall prove that the stationary wave of our quasilinear hyperbolic model is asymptotically stable. The weighted L2-energy method plays a crucial role in removing such a restriction on the sub-characteristic condition on the stationary wave.

Stability of the superposition of boundary layer and rarefaction wave for outflow problem on the two-fluid Navier–Stokes–Poisson system

This paper is concerned with the study of nonlinear stability of superposition of boundary layer and rarefaction wave on the two-fluid Navier–Stokes–Poisson system in the half line R+=:(0,+∞). On account of the quasineutral assumption and the absence of the electric field for the large time behavior, we successfully construct the boundary layer and rarefaction wave, and then we give the rigorous proofs of the stability theorems on the superposition of boundary layer and rarefaction wave under small perturbations for the corresponding initial boundary value problem of the Navier–Stokes–Poisson system, only provided the strength of boundary layer is small while the strength of rarefaction wave can be arbitrarily large. The complexity of nonlinear composite wave leads to many complicated terms in the course of establishing the a priori estimates. The proofs are given by an elementary L2 energy method.

Nonlinear stability of traveling wave solutions for the compressible fluid models of Korteweg type

This paper is concerned with the existence and time-asymptotic nonlinear stability of traveling wave solutions to the Cauchy problem of the one-dimensional compressible fluid models of Korteweg type, which governs the motions of the compressible fluids with internal capillarity. The existence of traveling wave solutions is obtained by the phase plane analysis, then the traveling wave solution is shown to be asymptotically stable by the elementary L2-energy method.

Asymptotic behavior of radially symmetric solutions for the Burgers equation in several space dimensions

We study large-time behavior of the radially symmetric solution for the Burgers equation on the exterior of a ball in a multi dimensional space, where boundary data at the far field are prescribed. Liu et al. (1998) considered the asymptotic behavior of the solution of scalar viscous conservation law for the case where the corresponding Riemann problem for the hyperbolic part admits a rarefaction wave. In the present paper, it is proved that for a radially symmetric solution to the Burgers equation on a multidimensional space, the asymptotic behaviors are the same as in Liu et al. (1998). Furthermore, we also derive the time convergence rate. The proof is given by a standard L2 energy method and a time weighted energy method.

Global existence and asymptotic decay of solutions to the nonlinear Timoshenko system with memory

In this paper we consider the initial-value problem for the nonlinear Timoshenko system with a memory term. Due to the regularity-loss property and weak dissipation, we have to assume stronger nonlinearity than usual. By virtue of the semi-group arguments, we obtain the global existence and optimal decay of solutions to the nonlinear problem under smallness and enough regularity assumptions on the initial data, where we employ a time-weighted L2 energy method combined with the optimal L2 decay of lower-order derivatives of solutions.

Asymptotic stability of strong rarefaction waves for the compressible fluid models of Korteweg type

This paper is concerned with the time-asymptotic behavior toward strong rarefaction waves of solutions to one-dimension compressible fluid models of Korteweg type, which governs the motions of the compressible fluids with internal capillarity. Assume that the corresponding Riemann problem to the compressible Euler system can be solved by rarefaction waves (VR,UR)(t,x). If the initial data is a small perturbation of an approximate rarefaction wave for (VR,UR)(t,x), we show that the corresponding Cauchy problem admits a unique global smooth solution which tends to (VR,UR)(t,x) time asymptotically. Since we do not require the strength of the rarefaction waves to be small, this result gives the nonlinear stability of strong rarefaction waves for the one-dimensional compressible fluid models of Korteweg type. The analysis is based on the elementary L2 energy method together with continuation argument.

GLOBAL EXISTENCE AND DECAY OF SOLUTIONS FOR A QUASI-LINEAR DISSIPATIVE PLATE EQUATION

In this paper we focus on the initial value problem of a quasi-linear dissipative plate equation with arbitrary spatial dimensions (n ≥ 1). This equation verifies the decay property of the regularity-loss type. To overcome the difficulty caused by the regularity-loss property, we employ a special time-weighted (with negative exponent) L2energy method combined with the optimal L2decay estimates of lower-order derivatives of solutions. We obtain the global existence and optimal decay estimates of solutions under smallness and enough regularity assumptions on the initial data. Moreover, we show that the solution can be approximated by a simple-looking function, which is the fundamental solution of the corresponding fourth-order linear parabolic equation.

Asymptotic Stability of Rarefaction Wave for the Navier–Stokes Equations for a Compressible Fluid in the Half Space

This paper is concerned with the asymptotic stability towards a rarefaction wave of the solution to an outflow problem for the Navier–Stokes equations in a compressible fluid in the Eulerian coordinate in the half space. This is the second one of our series of papers on this subject. In this paper, firstly we classify completely the time-asymptotic states, according to some parameters, that is the spatial-asymptotic states and boundary conditions, for this initial boundary value problem, and some pictures for the classification of time-asymptotic states are drawn in the state space. In order to prove the stability of the rarefaction wave, we use the solution to Burgers’ equation to construct a suitably smooth approximation of the rarefaction wave and establish some time-decay estimates in Lp-norm for the smoothed rarefaction wave. We then employ the L2-energy method to prove that the rarefaction wave is non-linearly stable under a small perturbation, as time goes to infinity.

DECAY PROPERTY OF REGULARITY-LOSS TYPE AND NONLINEAR EFFECTS FOR DISSIPATIVE TIMOSHENKO SYSTEM

We consider the initial value problem for a nonlinear version of the dissipative Timoshenko system. This syetem verifies the decay property of regularity-loss type. To overcome this difficulty caused by the regularity-loss property, we employ the time weighed L2 energy method which is combined with the optimal L2 decay estimates for lower order derivatives of solutions. Then we show the global existence and asymptotic decay of solutions under smallness and enough regularity conditions on the initial data. Moreover, we show that the solution approaches the linear diffusion wave expressed in terms of the superposition of the heat kernels as time tends to infinity.

ASYMPTOTIC DECAY TOWARD THE PLANAR RAREFACTION WAVES FOR A MODEL SYSTEM OF THE RADIATING GAS IN TWO DIMENSIONS

In this paper, we consider the asymptotic decay rate towards the planar rarefaction waves to the Cauchy problem for a hyperbolic–elliptic coupled system called as a model system of the radiating gas in two dimensions. The analysis based on the standard L2-energy method, L1-estimate and the monotonicity of profile obtained by the maximum principle.

THE NONLINEAR STABILITY OF TRAVELLING WAVE SOLUTIONS FOR A REACTING FLOW MODEL WITH SOURCE TERM

In this paper, author considers a 3 × 3 system for a reacting flow model propesed by [9]. Since this model has source term, it can be considered as a relaxation approximation to 2 × 2 systemsof conservation laws, which include the well-known p-system. From this viewpoint, the author establishes the global existence and the nonlinear stability of travelling wave solutions by L2 energy method.

ASYMPTOTIC DECAY TOWARD THE PLANAR RAREFACTION WAVES OF SOLUTIONS FOR VISCOUS CONSERVATION LAWS IN SEVERAL SPACE DIMENSIONS

This paper gives the asymptotic decay rate toward the planar rarefaction waves of the solutions for the scalar viscous conservation laws in two or more space dimensions. This is proved by a result on the decay rate of solutions for one-dimensional scalar viscous conservation laws and by using an L2-energy method with a weight of time.

Global solutions of the Neumann problem in one-dimensional nonlinear thermoelasticity

FOR THE equations of one-dimensional nonlinear thermoelasticity Slemrod [17] established in 1981 the global existence of smooth solutions for small initial data in the case of a bounded reference configuration for a homogeneous medium. The boundary conditions (traction free, constant temperature or rigidly clamped, insulated) he considered yield, through the differential equation, an additional boundary condition which was crucial for obtaining the necessary energy estimates. Subsequently, Jiang [7] discussed the same boundary conditions as in [17] in the case of an unbounded reference configuration (the half line). Kawashima [lo], Zheng and Shen [ 181, and Hrusa and Tarabek [6] dealt with the one-dimensional Cauchy problem. Recently, Racke and Shibata [13] proved a global existence theorem for the Dirichlet initial boundary value problem (rigidly clamped, constant temperature) in a bounded interval by using the decay estimates for the linearized equations and the L*-energy method, and the author [8] obtained a similar result for the problem in an unbounded interval (the half line) by only applying the L*-energy method. More recently, using a method similar to that in [13], Shibata [16] showed that global small solutions exist for the Neumann problem (traction free, insulated) in the case of a bounded reference configuration. For an unbounded reference configuration (the half line), the case of the Neumann boundary conditions remained open. Our purpose in this paper is to prove a global existence theorem for the Neumann problem in the half line and in a bounded interval for small data by applying the L2-energy method. The proof given here closely follows the pattern of the proof in [8], subject to the necessary modifications. We note that a globally defined classical solution should not be expected for large data. Indeed, Dafermos and Hsiao [4], and Hrusa and Messaoudi [5] showed that for specialized constitutive equations if the initial data are large, then the solution to the Cauchy problem will blow up in finite time. For three-dimensional nonlinear models, we refer the reader to the papers [3, 9, 10-12, 151 (and the references cited therein). This paper is organized as follows. In Section 1 the equations of one-dimensional nonlinear thermoelasticity are presented and the main theorem is stated. In Section 2 we give a local

THE ASYMPTOTIC BEHAVIOR OF SOLUTIONS OF INITIAL VALUE PROBLEM TO SYSTEM OF GAS DYNAMICS WITH VISCOSITY

This paper considers the system (A){υt-ux=ɛυxxut+p(υ)x=ɛuxx, x∈R, t>0where p(υ)=aυ-r, a>0, r>1,with (B)(υ(x, 0), u(x, 0))=(υ0(x), u0(x)), x∈Rwhere x→±∞ lim(υ0(x), u0(x))=(υ±, u±)provided (C1)0<υ-<υ+, u+<u-(C2)0<υ+<υ-, u+<u-or (C3)0<υ+<υ-, u-<u+(C4)0<υ-<υ+, u-<u+The traveling wave solution of problem (A), (B) is proved to be asymptotically stable when (C)1, (C)2 are satisfied and rarefaction wave solution is proved to be asymptotical when (C)3, (C)4 are satisfied, provided that the initial disturbance is suitably small and of zero constant component. The proof is given by the elemental L2 energy method.